Standard +0.3 This is a straightforward implicit differentiation question requiring students to differentiate implicitly using the product rule, substitute a given point to find the gradient, then find the perpendicular gradient and write the normal equation. While it involves multiple steps (differentiate, substitute, find normal gradient, form equation), each step is standard technique with no novel insight required. The 8 marks reflect the working steps rather than conceptual difficulty, making this slightly easier than average.
The equation of a curve is \(3x^2 + 4xy + y^2 = 24\). Find the equation of the normal to the curve at the point \((1, 3)\), giving your answer in the form \(ax + by + c = 0\) where \(a\), \(b\) and \(c\) are integers. [8]
Equate attempt of derivative of left-hand side to zero
Substitute (1,3) to find numerical value of derivative
Obtain −18 or −9
10 5
Obtain 10 or 5 as gradient of normal, following their numerical value of derivative
18 9
Form equation of normal at (1,3)
Answer
Marks
Obtain 5x−9y+22=0 or equivalent of requested form
B1
B1
M1
M1
A1
A1
M1
Answer
Marks
A1
[8]
7 (i)
(ii) (a)
Answer
Marks
(b)
Substitute x=−3, equate to zero and obtain 27a+3b=39 or equivalent
Substitute x=−2 and equate to 18
Obtain 8a+2b=6 or equivalent
Solve a relevant pair of linear equations for a and b
Obtain a=2 and b=−5
Attempt division by x+3 at least as far as 2x2 +kx
Obtain quotient 2x2 −3x+4
Calculate discriminant of 3-term quadratic expression, or equivalent
Obtain −23 and conclude appropriately
State cosy=−1
3
Obtain 109.5, dependent *B
Answer
Marks
Obtain –109.5 and no others between –180 and 180, dependent *B
B1
M1
A1
M1
A1
M1
A1
M1
A1
*B1
B1
Answer
Marks
DB1
[5]
[4]
[3]
Question 6:
6 | dy
Differentiate 4xy to obtain 4y+4x
dx
dy
Differentiate y2 to obtain 2y
dx
Equate attempt of derivative of left-hand side to zero
Substitute (1,3) to find numerical value of derivative
Obtain −18 or −9
10 5
Obtain 10 or 5 as gradient of normal, following their numerical value of derivative
18 9
Form equation of normal at (1,3)
Obtain 5x−9y+22=0 or equivalent of requested form | B1
B1
M1
M1
A1
A1
M1
A1 | [8]
7 (i)
(ii) (a)
(b) | Substitute x=−3, equate to zero and obtain 27a+3b=39 or equivalent
Substitute x=−2 and equate to 18
Obtain 8a+2b=6 or equivalent
Solve a relevant pair of linear equations for a and b
Obtain a=2 and b=−5
Attempt division by x+3 at least as far as 2x2 +kx
Obtain quotient 2x2 −3x+4
Calculate discriminant of 3-term quadratic expression, or equivalent
Obtain −23 and conclude appropriately
State cosy=−1
3
Obtain 109.5, dependent *B
Obtain –109.5 and no others between –180 and 180, dependent *B | B1
M1
A1
M1
A1
M1
A1
M1
A1
*B1
B1
DB1 | [5]
[4]
[3]
The equation of a curve is $3x^2 + 4xy + y^2 = 24$. Find the equation of the normal to the curve at the point $(1, 3)$, giving your answer in the form $ax + by + c = 0$ where $a$, $b$ and $c$ are integers. [8]
\hfill \mbox{\textit{CAIE P2 2016 Q6 [8]}}